1. 3. Differential operators
Gradient
1- Definition. f(x,y,z) is a differentiable scalar field
2 – Physical meaning: is the local variation of f along dr.
Particularly, grad f is perpendicular to the line f = ctt.
Summer School 2007
B. Rossetto

2. 3. Differential operators
Divergence
1 – Definition
is a differentiable vector field x
x+dx
2 – Physical meaning
is associated to local conservation laws: for example,
we’ll show that if the mass of fluid (or of charge) outcoming from a domain is equal to the mass entering, then
is the fluid velocity (or the current) vectorfield
Summer School 2007
B. Rossetto

3. 3. Differential operators
Curl
1 – Definition.
is a differentiable vector field
2 – Physical meaning: is related to
the local rotation of the vectorfield: If
is the fluid velocity vectorfield
Summer School 2007
B. Rossetto

4. 3. Differential operators
Laplacian: definitions
1 – Scalar Laplacian. f(x,y,z) is a differentiable scalar field
2 – Vector Laplacian.
is a differentiable vector field
Summer School 2007
B. Rossetto

5. 3. Differential operators
Laplacian: physical meaning
f(x)
As a second derivative, the one-dimensional
Laplacian operator is related to minima and
maxima: when the second derivative is
positive (negative), the curvature is
concave (convexe).
convex
concave x
In most of situations, the 2-dimensional
Laplacian operator is also related to local
minima and maxima. If vE is positive: E
Summer School 2007
B. Rossetto

6. 3. Differential operators
Summary
Operator
grad
div
curl
Laplacian is
a vector
a scalar
(resp. a vector)
concerns
a scalar field
a vector field
(resp. a vector field)
Definition
resp.
Summer School 2007
B. Rossetto

7. 3. Differential operators
Cylindrical coordinates
Summer School 2007
B. Rossetto

8. 3. Differential operators
Cylindrical coordinates
Summer School 2007
B. Rossetto

9. 3. Differential operators
Spherical coordinates
Summer School 2007
B. Rossetto

10. 3. Differential operators
Conservative vectorfield
Theorem. If there exists f such that
then H P
Consequently, the value of the integral doesn’t depend
on the path, but only on its beginning A and its end B.
We say that the vectorfield is conservative
Proof.
Summer School 2007
B. Rossetto

11. 3. Differential operators
1st Stokes formula: vectorfield global circulation
Theorem. If S(C) is any oriented surface delimited by C:
S(C) C
Sketch of proof. y Vy . Vx . . P x .
… and then extend to any surface delimited by C.
Summer School 2007
B. Rossetto

12. 3. Differential operators
2nd Stokes formula: global conservation laws
Theorem. If V(C) is the volume delimited by S
Sketch of proof. Flow through the oriented
elementary planes x = ctt and x+dx = ctt : x
x+dx
-Vx(x,y,z).dydz + Vx (x+dx,y,z).dydz
and then extend this expression to the lateral surface of the cube.
Other expression:
extended to the vol. of the elementary cube:
Summer School 2007
B. Rossetto

13. 3. Differential operators
Vector identities
Use Einstein convention and Levi-Civita symbol to show them
curl(grad f) = 0
Summer School 2007
B. Rossetto